Next: Why is it working Up: Guitton: Signal/noise separation Previous: Introduction

The subtraction method

In this section, I show that starting from the least-squares inverse of the subtraction method, I can retrieve a fitting goal that resembles the fitting goal of the filtering technique. I first assume that the data is the sum of the signal and the noise as follows:
 (1)
Now I assume that I have two modeling operators and for the signal and noise respectively such that
 (2)
where and are unknown model parameters. In this paper I assume that the two modeling operators are orthogonal, meaning that they model different regions of the data space. Inserting the modeling operators into equation (1) I have
 (3)
I then want to estimate and so that
 (4)
In a least-squares sense, I have to minimize the following objective function:
 (5)
The least-squares inverse of the model parameters is given by Guitton (2001)   with
 (6)
One can easily recognize in the definition of the identity matrix minus the signal resolution operator and for , the identity matrix minus the noise resolution operator. Therefore, and are signal and noise filtering operators respectively with
 (7)
An interesting property of and is that and : they are called projectors. It is easy to verify that they are Hermitian operators as well Guitton (2001). Now, looking at the first row in equation(6), I have
 (8)
This expression is the least-squares inverse for the following objective function
 (9)
which can be rewritten
 (10)
or
 (11)
using the properties of . Therefore, equation (12) is the objective function for the fitting goal
 (12)
Similarly, if I look at the second row of equation (6), I end up with
 (13)

Next: Why is it working Up: Guitton: Signal/noise separation Previous: Introduction
Stanford Exploration Project
7/8/2003